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Properties

Label 18.0.69165347071...1307.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 7^{12}\cdot 13^{12}\cdot 109^{12}$
Root discriminant $2398.80$
Ramified primes $3, 7, 13, 109$
Class number $77165233200$ (GRH)
Class group $[3, 3, 3, 3, 3, 6, 18, 990, 2970]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7625597484987, 0, 0, -3842823830391, 0, 0, 2032589480652, 0, 0, 25079348385, 0, 0, 103266244, 0, 0, -9919, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9919*x^15 + 103266244*x^12 + 25079348385*x^9 + 2032589480652*x^6 - 3842823830391*x^3 + 7625597484987)
 
gp: K = bnfinit(x^18 - 9919*x^15 + 103266244*x^12 + 25079348385*x^9 + 2032589480652*x^6 - 3842823830391*x^3 + 7625597484987, 1)
 

Normalized defining polynomial

\( x^{18} - 9919 x^{15} + 103266244 x^{12} + 25079348385 x^{9} + 2032589480652 x^{6} - 3842823830391 x^{3} + 7625597484987 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-6916534707190682077071725436128319105898712657538875913221307=-\,3^{27}\cdot 7^{12}\cdot 13^{12}\cdot 109^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2398.80$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 7, 13, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{22} a^{6} + \frac{1}{22} a^{3} - \frac{7}{22}$, $\frac{1}{66} a^{7} + \frac{23}{66} a^{4} - \frac{29}{66} a$, $\frac{1}{594} a^{8} + \frac{287}{594} a^{5} + \frac{235}{594} a^{2}$, $\frac{1}{1782} a^{9} - \frac{37}{1782} a^{6} - \frac{89}{1782} a^{3} + \frac{3}{11}$, $\frac{1}{16038} a^{10} + \frac{2}{729} a^{7} - \frac{4}{8019} a^{4} - \frac{5}{22} a$, $\frac{1}{48114} a^{11} - \frac{37}{48114} a^{8} + \frac{8821}{48114} a^{5} - \frac{41}{297} a^{2}$, $\frac{1}{100667286324} a^{12} + \frac{8126185}{50333643162} a^{9} - \frac{1119122171}{100667286324} a^{6} - \frac{4986895}{11507463} a^{3} - \frac{2537531}{5114428}$, $\frac{1}{2718016730748} a^{13} + \frac{18185888}{679504182687} a^{10} - \frac{12360867389}{2718016730748} a^{7} - \frac{1983148603}{5592627018} a^{4} + \frac{68599513}{138089556} a$, $\frac{1}{73386451730196} a^{14} - \frac{151287658}{18346612932549} a^{11} - \frac{42188211485}{73386451730196} a^{8} + \frac{53954280329}{151000929486} a^{5} + \frac{1851210145}{3728418012} a^{2}$, $\frac{1}{214362039131863502600556} a^{15} + \frac{64005122792}{53590509782965875650139} a^{12} + \frac{38497386102965415613}{214362039131863502600556} a^{9} + \frac{2012847113532010397}{441074154592311733746} a^{6} + \frac{4247284411369354645}{10890719866476832932} a^{3} - \frac{951326703447}{2286387982262}$, $\frac{1}{5787775056560314570215012} a^{16} + \frac{64005122792}{1446943764140078642553753} a^{13} + \frac{38497386102965415613}{5787775056560314570215012} a^{10} + \frac{82208147948497780169}{11909002173992416811142} a^{7} - \frac{124461223101538670915}{294049436394874489164} a^{4} + \frac{83277313534825}{679057230731814} a$, $\frac{1}{156269926527128493395805324} a^{17} + \frac{64005122792}{39067481631782123348951331} a^{14} + \frac{38497386102965415613}{156269926527128493395805324} a^{11} + \frac{82208147948497780169}{321543058697795253900834} a^{8} - \frac{418510659496413160079}{7939334782661611207428} a^{5} - \frac{3312008840124245}{18334545229758978} a^{2}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{6}\times C_{18}\times C_{990}\times C_{2970}$, which has order $77165233200$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{110674342}{442896775065833683059} a^{15} - \frac{1097974033975}{442896775065833683059} a^{12} + \frac{11430909140143078}{442896775065833683059} a^{9} + \frac{11337404655348226}{1822620473521949313} a^{6} + \frac{11441311235996338}{22501487327431473} a^{3} + \frac{35247270609}{1143193991131} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 78066072406331.89 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.295159683.1 x3, 3.3.47155689.4, 6.0.261357715405981467.1, 6.0.6670977015194163.2, Deg 6 x2, 9.3.1518391112020733837237059617363.1 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 sibling: data not computed
Degree 9 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
$13$13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
$109$109.3.2.3$x^{3} - 3924$$3$$1$$2$$C_3$$[\ ]_{3}$
109.3.2.3$x^{3} - 3924$$3$$1$$2$$C_3$$[\ ]_{3}$
109.3.2.3$x^{3} - 3924$$3$$1$$2$$C_3$$[\ ]_{3}$
109.3.2.3$x^{3} - 3924$$3$$1$$2$$C_3$$[\ ]_{3}$
109.3.2.3$x^{3} - 3924$$3$$1$$2$$C_3$$[\ ]_{3}$
109.3.2.3$x^{3} - 3924$$3$$1$$2$$C_3$$[\ ]_{3}$